volume 4, issue 4, article 70, 2003.
Received 15 September, 2003;
accepted 17 September, 2003.
Communicated by:Th. M. Rassias
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Journal of Inequalities in Pure and Applied Mathematics
SOME INEQUALITIES ASSOCIATED WITH A LINEAR OPERATOR DEFINED FOR A CLASS OF MULTIVALENT FUNCTIONS
V. RAVICHANDRAN, N. SEENIVASAGAN AND H.M. SRIVASTAVA
Department of Computer Applications Sri Venkateswara College of Engineering Sriperumbudur 602105, Tamil Nadu, India.
EMail:vravi@svce.ac.in Department of Mathematics Sindhi College
123 P.H. Road, Numbal
Chennai 600077, Tamil Nadu, India.
EMail:vasagan2000@yahoo.co.in Department of Mathematics and Statistics University of Victoria
Victoria, British Columbia V8W 3P4 Canada.
EMail:harimsri@math.uvic.ca
URL:http://www.math.uvic.ca/faculty/harimsri/
c
2000Victoria University ISSN (electronic): 1443-5756 122-03
Some Inequalities Associated with a Linear Operator Defined
for a Class of Multivalent Functions
V. Ravichandran, N. Seenivasagan and
H.M. Srivastava
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Abstract
The authors derive several inequalities associated with differential subordina- tions between analytic functions and a linear operator defined for a certain family ofp-valent functions, which is introduced here by means of this linear operator. Some special cases and consequences of the main results are also considered.
2000 Mathematics Subject Classification:Primary 30C45.
Key words: Analytic functions, Univalent and multivalent functions, Differential sub- ordination, Schwarz function, Ruscheweyh derivatives, Hadamard prod- uct (or convolution), Linear operator, Convex functions, Starlike func- tions.
The present investigation was supported, in part, by the Natural Sciences and Engi- neering Research Council of Canada under Grant OGP0007353.
Contents
1 Introduction, Definitions and Preliminaries . . . 3 2 Inequalities Involving the Linear OperatorLp(a, c) . . . 11 3 Further Results Involving Differential Subordination Between
Analytic Functions . . . 16 References
Some Inequalities Associated with a Linear Operator Defined
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1. Introduction, Definitions and Preliminaries
LetA(p, n)denote the class of functionsf normalized by (1.1) f(z) = zp+
∞
X
k=p+n
akzk (p, n∈N:={1,2,3, . . .}), which are analytic in the open unit disk
U:={z :z ∈C and |z|<1}. In particular, we set
A(p,1) =:Ap and A(1,1) =:A =A1.
A function f∈ A(p, n) is said to be in the class A(p, n;α) if it satisfies the following inequality:
(1.2) R
1 + zf00(z) f0(z)
< α (z ∈U;α > p).
We also denote by K(α) and S∗(α), respectively, the usual subclasses of A consisting of functions which are convex of order α inUand starlike of order αinU. Thus we have (see, for details, [3] and [9])
(1.3) K(α) :=
f:f ∈ A andR
1 + zf00(z) f0(z)
> α (z ∈U; 05α <1)
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and
(1.4) S∗(α) :=
f:f ∈ AandR
zf0(z) f(z)
> α (z∈U; 05α <1)
. In particular, we write
K(0) =:K and S∗(0) =:S∗.
For the above-defined classA(p, n;α)ofp-valent functions, Owa et al. [5]
proved the following results.
Theorem A. (Owa et al. [5, p. 8, Theorem 1]). If f(z)∈ A(p, n;α)
p < α5p+1 2n
,
then
(1.5) R
f(z) zf0(z)
> 2p+n
(2α+n)p (z ∈U).
Theorem B. (Owa et al. [5, p. 10, Theorem 2]). If f(z)∈ A(p, n;α)
p < α5p+1 2n
,
then
(1.6) 0<R
zf0(z) f(z)
< (2α+n)p
2p+n (z ∈U).
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In fact, as already observed by Owa et al. [5, p. 10], various further special cases of (for example) TheoremBwhenp=n = 1were considered earlier by Nunokawa [4], Saitoh et al. [7], and Singh and Singh [8].
The main object of this paper is to present an extension of each of the inequalities (1.5) and (1.6) asserted by Theorem A and Theorem B, respec- tively, to hold true for a linear operator associated with a certain general class A(p, n;a, c, α)ofp-valent functions, which we introduce here.
For two functionsf(z)given by (1.1) andg(z)given by g(z) =zp +
∞
X
k=p+n
bk zk (p, n∈N),
the Hadamard product (or convolution)(f∗g) (z)is defined, as usual, by (1.7) (f ∗g) (z) :=zp+
∞
X
k=p+n
akbkzk =: (g∗f) (z).
In terms of the Pochhammer symbol(λ)kor the shifted factorial, since (1)k=k! (k ∈N0 :=N∪ {0}),
given by
(λ)0 := 1 and (λ)k :=λ(λ+ 1)· · ·(λ+k−1) (k ∈N), we now define the functionφp(a, c;z)by
(1.8) φp(a, c;z) := zp+
∞
X
k=1
(a)k (c)k zk+p
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z ∈U; a ∈R; c∈R\Z−0; Z−0 :={0,−1,−2, . . .}
.
Corresponding to the functionφp(a, c;z), Saitoh [6] introduced a linear opera- torLp(a, c)which is defined by means of the following Hadamard product (or convolution):
(1.9) Lp(a, c)f(z) := φp(a, c;z)∗f(z) (f ∈ Ap) or, equivalently, by
(1.10) Lp(a, c)f(z) :=zp+
∞
X
k=1
(a)k
(c)k ak+p zk+p (z ∈U).
The definition (1.9) or (1.10) of the linear operator Lp(a, c) is motivated essentially by the familiar Carlson-Shaffer operator
L(a, c) := L1(a, c),
which has been used widely on such spaces of analytic and univalent functions in U asK(α) and S∗(α)defined by (1.3) and (1.4), respectively (see, for ex- ample, [9]). A linear operatorLp(a, c), analogous toLp(a, c)considered here, was investigated recently by Liu and Srivastava [2] on the space of meromorphi- callyp-valent functions in U. We remark in passing that a much more general convolution operator than the operatorLp(a, c)considered here, involving the generalized hypergeometric function in the defining Hadamard product (or con- volution), was introduced earlier by Dziok and Srivastava [1].
Making use of the linear operatorLp(a, c)defined by (1.9) or (1.10), we say that a functionf ∈ A(p, n)is in the aforementioned general classA(p, n;a, c, α)
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if it satisfies the following inequality:
(1.11) R
Lp(a+ 2, c)f(z) Lp(a+ 1, c)f(z)
< α z ∈U; α >1; a∈R; c∈R\Z−0
.
The Ruscheweyh derivative off(z)of orderδ+p−1is defined by (1.12) Dδ+p−1 f(z) := zp
(1−z)δ+p ∗f(z) (f∈ A(p, n) ; δ ∈R\(−∞,−p]) or, equivalently, by
(1.13) Dδ+p−1 f(z) :=zp +
∞
X
k=p+n
δ+k−1 k−p
ak zk
(f ∈ A(p, n) ; δ ∈R\(−∞,−p]).
In particular, if δ = l(l+p∈N), we find from the definition (1.12) or (1.13) that
(1.14) Dl+p−1 f(z) = zp (l+p−1)!
dl+p−1 dzl+p−1
zl−1 f(z) , (f ∈ A(p, n) ; l+p∈N).
Since
(1.15) Lp(δ+p,1)f(z) =Dδ+p−1 f(z),
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(f ∈ A(p, n) ; δ ∈R\(−∞,−p]),
which can easily be verified by comparing the definitions (1.10) and (1.13), we may set
(1.16) A(p, n;δ+p,1, α) =:A(p, n;δ, α).
Thus a function f ∈ A(p, n)is in the classA(p, n;δ, α)if it satisfies the fol- lowing inequality:
(1.17) R
Dδ+p+1 f(z) Dδ+p f(z)
< α, (z ∈U; α >1; δ∈R\(−∞,−p]).
Finally, for two functions f and g analytic in U, we say that the function f(z)is subordinate tog(z)inU, and write
f ≺g or f(z)≺g(z) (z ∈U), if there exists a Schwarz functionw(z), analytic inUwith
w(0) = 0 and |w(z)|<1 (z ∈U), such that
(1.18) f(z) =g w(z)
(z ∈U).
In particular, if the function g is univalent in U, the above subordination is equivalent to
f(0) =g(0) and f(U)⊂g(U).
In our present investigation of the above-defined general classA(p, n;a, c, α), we shall require each of the following lemmas.
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Lemma 1. (cf. Miller and Mocanu [3, p. 35, Theorem 2.3i (i)]). Let Ω be a set in the complex plane Cand suppose that Φ (u, v;z)is a complex-valued mapping:
Φ :C2×U→C, where
u=u1+iu2 and v =v1+iv2.
Also letΦ (iu2, v1;z)∈/ Ω for allz ∈Uand for all realu2 andv1 such that
(1.19) v1 5−1
2n 1 +u22 . If
q(z) = 1 +cnzn+cn+1zn+1+· · · is analytic inUand
Φ (q(z), zq0(z) ;z)∈Ω (z ∈U), then
R{q(z)}>0 (z ∈U).
Lemma 2. (cf. Miller and Mocanu [3, p. 132, Theorem 3.4h]). Letψ(z)be univalent inUand suppose that the functionsϑandϕare analytic in a domain D ⊃ ψ(U) with ϕ(ζ) 6= 0 when ζ ∈ ψ(U). Define the functions Q(z) and h(z)by
(1.20) Q(z) := zψ0(z)ϕ ψ(z)
and h(z) :=ϑ ψ(z)
+Q(z),
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and assume that
(i) Q(z)is starlike univalent inU and
(ii) R
zh0(z) Q(z)
>0 (z ∈U). If
(1.21) ϑ
q(z)
+zq0(z)ϕ q(z)
≺h(z) (z ∈U), then
q(z)≺ψ(z) (z ∈U) andψ(z)is the best dominant.
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2. Inequalities Involving the Linear Operator L
p(a, c)
By appealing to Lemma 1 of the preceding section, we first prove Theorem1 below.
Theorem 1. Let the parametersaandαsatisfy the following inequalities:
(2.1) a >−1 and 1< α51 + n 2(a+ 1). Iff(z)∈ A(p, n;a, c, α), then
(2.2) R
Lp(a, c)f(z) Lp(a+ 1, c)f(z)
> 2a+n
2α(a+ 1)−2 +n (z∈U) and
(2.3) R
Lp(a+ 1, c)f(z) Lp(a, c)f(z)
< 2α(a+ 1)−2 +n
2a+n (z ∈U).
Proof. Define the functionq(z)by
(2.4) (1−β)q(z) +β = Lp(a, c)f(z)
Lp(a+ 1, c)f(z) (z∈U), where
(2.5) β := 2a+n
2α(a+ 1)−2 +n.
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Then, clearly,q(z)is analytic inUand
q(z) = 1 +cn zn+cn+1 zn+1+· · · (z ∈U).
By a simple computation, we observe from (2.4) that (2.6) (1−β)zq0(z)
(1−β)q(z) +β = z Lp(a, c)f(z)0
Lp(a, c)f(z) − z Lp(a+ 1, c)f(z)0 Lp(a+ 1, c)f(z) . Making use of the familiar identity:
(2.7) z Lp(a, c)f(z)0
=aLp(a+ 1, c)f(z)−(a−p)Lp(a, c)f(z), we find from (2.6) that
(1−β)zq0(z)
(1−β)q(z) +β = 1 +a Lp(a+ 1, c)f(z)
Lp(a, c)f(z) −(a+ 1) Lp(a+ 2, c)f(z) Lp(a+ 1, c)f(z), which, in view of (2.4), yields
Lp(a+ 2, c)f(z)
Lp(a+ 1, c)f(z) = 1
a+ 1 + 1 a+ 1
a
(1−β)q(z) +β − (1−β)zq0(z) (1−β)q(z) +β
or, equivalently,
(2.8) Lp(a+ 2, c)f(z)
Lp(a+ 1, c)f(z) = 1 a+ 1
1 + a−(1−β)zq0(z) (1−β)q(z) +β
.
If we defineΦ(u, v;z)by
(2.9) Φ(u, v;z) := 1
a+ 1
1 + a−(1−β)v (1−β)u+β
,
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then, by the hypothesis of Theorem1thatf ∈ A(p, n;a, c, α), we have R{Φ (q(z), zq0(z);z)}=R
Lp(a+ 2, c)f(z) Lp(a+ 1, c)f(z)
< α (z ∈U; α >1).
We will now show that
R{Φ (iu2, v1;z)}=α
for all z ∈ U and for all real u2 and v1 constrained by the inequality (1.19).
Indeed we find from (2.9) that R{Φ (iu2, v1;z)}= 1
a+ 1
1 +R
a−(1−β)v1 (1−β)iu2+β
= 1
a+ 1
1 +R
[a−(1−β)v1][β−(1−β)iu2] (1−β)2u22+β2
= 1
a+ 1
1 + [a−(1−β)v1]β (1−β)2u22+β2
,
so that, by using (1.19), we have (2.10) R{Φ (iu2, v1;z)}= 1
a+ 1
1 + β[a+12n(1−β)(1 +u22)]
(1−β)2u22+β2
(z∈U). From the inequalities in (2.1), we get
n 2β2 =
a+1
2n(1−β)
(1−β),
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and hence the function
a+ 12n(1−β)(1 +x2) (1−β)2x2+β2
is an increasing function forx=0. Thus we find from (2.10) that R{Φ (iu2, v1;z)}= 1
a+ 1
1 + a+ 12n(1−β) β
=α (z ∈U). The first assertion (2.2) of Theorem1follows by applying Lemma1.
Next, we define the functionψ(z)by ψ(z) := Lp(a, c)f(z)
Lp(a+ 1, c)f(z) (z ∈U),
where β is given by (2.5). Then, in view of the already proven assertion (2.2) of Theorem1, we have
(2.11) R{ψ(z)}> β > 0 (z ∈U) so that
(2.12) R
1 ψ(z)
>0 (z ∈U).
Since (2.12) holds true, we have R{ψ(z)}R
1 ψ(z)
5|ψ(z)| · 1
|ψ(z)| = 1,
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or
R 1
ψ(z)
5 1
R{ψ(z)} (z ∈U), which, in view of (2.11), yields
0<R 1
ψ(z)
< 1
β (z ∈U) which is the second assertion (2.3) of Theorem1.
The following result is a special case of Theorem1obtained by taking a=δ+p and c= 1.
Corollary 1. If
f(z)∈ A(p, n;δ, α)
δ+p > 1; 15α <1 + n 2(δ+p+ 1)
,
then
R
Dδ+p−1f(z) Dδ+pf(z)
> 2δ+ 2p+n
2α(δ+p+ 1)−2 +n (z ∈U), and
R
Dδ+pf(z) Dδ+p−1f(z)
< 2α(δ+p+ 1)−2 +n
2δ+ 2p+n (z ∈U).
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3. Further Results Involving Differential Subordi- nation Between Analytic Functions
We begin by proving the following result.
Lemma 3. Let the functionsq(z)andψ(z)be analytic inUand suppose that ψ(z)6= 0 (z ∈U)
is also univalent inUand thatzψ0(z)/ψ(z)is starlike univalent inU. If
(3.1) R
α β
1 ψ(z)+
1 + zψ00(z)
ψ0(z) −zψ0(z) ψ(z)
>0, (z ∈U; α, β ∈C; β 6= 0)
and
(3.2) α
q(z)−β zq0(z) q(z) ≺ α
ψ(z) −β zψ0(z) ψ(z) , (z ∈U; α, β ∈C; β 6= 0), then
q(z)≺ψ(z) (z ∈U) andq(z)is the best dominant.
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Proof. By setting
ϑ(ζ) = α
ζ and ϕ(ζ) = −β ζ,
it is easily observed that bothϑ(ζ)andϕ(ζ)are analytic inC\{0}and that ϕ(ζ)6= 0 (ζ ∈C\ {0}).
Also, by letting
(3.3) Q(z) =zψ0(z)ϕ ψ(z)
=−β zψ0(z) ψ(z)
and
(3.4) h(z) =ϑ ψ(z)
+Q(z) = α
ψ(z) −β zψ0(z) ψ(z) ,
we find thatQ(z)is starlike univalent inUand that R
zh0(z) Q(z)
=R α
β 1 ψ(z)+
1 + zψ00(z)
ψ0(z) −zψ0(z) ψ(z)
>0, (z ∈U; α, β ∈C; β 6= 0),
by the hypothesis (3.1) of Lemma3. Thus, by applying Lemma2, our proof of Lemma3is completed.
We now prove the following result involving differential subordination be- tween analytic functions.
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Theorem 2. Let the functionψ(z)6= 0 (z ∈U)be analytic and univalent inU and suppose thatzψ0(z)/ψ(z)is starlike univalent inUand
(3.5) R
a ψ(z)+
1 + zψ00(z)
ψ0(z) −zψ0(z) ψ(z)
>0 (z ∈U; a ∈C\ {−1}).
Iff ∈ Apsatisfies the following subordination:
(3.6) Lp(a+ 2, c)f(z)
Lp(a+ 1, c)f(z) ≺ 1 a+ 1
1 + a−zψ0(z) ψ(z)
(z∈U), then
(3.7) Lp(a, c)f(z)
Lp(a+ 1, c)f(z) ≺ψ(z) (z ∈U) andψ(z)is the best dominant.
Proof. Let the functionq(z)be defined by q(z) := Lp(a, c)f(z)
Lp(a+ 1, c)f(z) (z ∈U; f ∈ Ap), so that, by a straightforward computation, we have
(3.8) zq0(z)
q(z) = z Lp(a, c)f(z)0
Lp(a, c)f(z) − z Lp(a+ 1, c)f(z)0 Lp(a+ 1, c)f(z) ,
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which follows also from (2.6) in the special case whenβ = 0.
Making use of the familiar identity (2.7) once again (or directly from (2.8) withβ = 0), we find that
Lp(a+ 2, c)f(z)
Lp(a+ 1, c)f(z) =a Lp(a+ 1, c)f(z)
Lp(a, c)f(z) −(a+ 1) Lp(a+ 2, c)f(z) Lp(a+ 1, c)f(z) + 1
= 1
a+ 1
1 + a
q(z)− zq0(z) q(z)
,
which, in light of the hypothesis (3.6) of Theorem2, yields the following sub- ordination:
a
q(z) − zq0(z) q(z) ≺ a
ψ(z) − zψ0(z)
ψ(z) (z ∈U).
The assertion (3.7) of Theorem2now follows from Lemma3.
Remark 1. If the functionψ(z)is such that
R{ψ(z)}>0 (z ∈U)
and if zψ0(z)/ψ(z) is starlike in U, then the condition (3.5) is satisfied for a >0.
In its special case when
a=δ+p and c= 1, Theorem2yields the following result.
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Corollary 2. Let the functionψ(z)6= 0 (z ∈U)be analytic and univalent inU and suppose thatzψ0(z)/ψ(z)is starlike univalent inUand
R
δ+p ψ(z) +
1 + zψ00(z)
ψ0(z) − zψ0(z) ψ(z)
>0 (z ∈U; δ∈R\(−∞, p]).
Iff ∈ Asatisfies the following subordination:
Dδ+p+1f(z)
Dδ+pf(z) ≺ 1 δ+p+ 1
1 + δ+p−zψ0(z) ψ(z)
(z ∈U), then
Dδ+p−1f(z)
Dδ+pf(z) ≺ψ(z) (z ∈U).
Lastly, by using a similar technique as above, we can prove Theorem3be- low.
Theorem 3. Iff ∈ A(p, n)and (3.9) 1 + zf00(z)
f0(z) ≺p 1 +Bzn
1 +Azn − n(A−B)zn (1 +Azn)(1 +Bzn), (z ∈U; −15B < A51),
then
(3.10) pf(z)
zf0(z) ≺ 1 +Azn
1 +Bzn (z ∈U).
Some Inequalities Associated with a Linear Operator Defined
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Proof. Let the functionq(z)be defined by
(3.11) q(z) := pf(z)
zf0(z) (z∈U; f ∈ A(p, n)), so that
(3.12) 1 + zf00(z)
f0(z) = p
q(z) −zq0(z) q(z) .
If the functionψ(z)is defined by ψ(z) := 1 +Azn
1 +Bzn (−15B < A51; z ∈U), then, in view of (3.9) and (3.12), we get
p
q(z) − zq0(z) q(z) ≺ p
ψ(z) − zψ0(z)
ψ(z) (z ∈U).
The result (Theorem3) now follows from Lemma3(withα =pand β = 1).
The following result is a simple consequence of Theorem3.
Corollary 3. Iff ∈ Asatisfies the following subordination:
1 + zf00(z)
f0(z) ≺ 1−4z+z2
1−z2 (z ∈U),
Some Inequalities Associated with a Linear Operator Defined
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then
(3.13) R
f(z) zf0(z)
>0 (z ∈U) or, equivalently,f is starlike inU(that is,f ∈ S∗).
Remark 2. The foregoing analysis can be applied mutatis mutandis in order to rederive TheoremAof Owa et al. [5]. Indeed, if
(3.14) f(z)∈ A(p, n;α)
p < α5p+1 2n
,
then we can first show that
1 + zf00(z)
f0(z) ≺ψ(z) (z ∈U), where
ψ(z) :=p 1 +Bzn
1 +Azn − n(A−B)zn
(1 +Azn)(1 +Bzn) = p(1 +Bzn)2−n(A+ 1)zn (1 +Azn)(1−zn)
A= 1−2β; B =−1; β= 2p+n 2α+n
.
By letting
u(θ) :=R{ψ(z)} z =eiθ/n∈∂U; 05θ52nπ ,
Some Inequalities Associated with a Linear Operator Defined
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it is easily seen for
u(θ) = (1−A) [2p+n(1 +A)−2pcosθ]
2(1 +A2+ 2Acosθ) (05θ52nπ) that
(3.15) u(θ)=u(π) =(1−A) [2p+n(1 +A) + 2p]
2(1−A)2 =α (05θ52nπ), which shows thatq(U)contains the half-planeR(w)5α, whereq(z)is given, as before, by(3.11). Thus, under the hypothesis(3.14), we have the subordina- tion(3.9)and hence(by Theorem3)also the subordination(3.10), which leads us to the assertion(1.5)of TheoremA.
Some Inequalities Associated with a Linear Operator Defined
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References
[1] J. DZIOK AND H.M. SRIVASTAVA, Classes of analytic functions associ- ated with the generalized hypergeometric function, Appl. Math. Comput., 103 (1999), 1–13.
[2] J.-L. LIU ANDH.M. SRIVASTAVA, A linear operator and associated fam- ilies of meromorphically multivalent functions, J. Math. Anal. Appl., 259 (2001), 566–581.
[3] S.S. MILLER AND P.T. MOCANU, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Ap- plied Mathematics (No. 225), Marcel Dekker, New York and Basel, 2000.
[4] M. NUNOKAWA, A sufficient condition for univalence and starlikeness, Proc. Japan Acad. Ser. A Math. Sci., 65 (1989), 163–164.
[5] S. OWA, M. NUNOKAWA AND H.M. SRIVASTAVA, A certain class of multivalent functions, Appl. Math. Lett., 10 (2) (1997), 7–10.
[6] H. SAITOH, A linear operator and its applications of first order differential subordinations, Math. Japon., 44 (1996), 31–38.
[7] H. SAITOH, M. NUNOKAWA, S. FUKUI AND S. OWA, A remark on close-to-convex and starlike functions, Bull. Soc. Roy. Sci. Liège, 57 (1988), 137–141.
[8] R. SINGH AND S. SINGH, Some sufficient conditions for univalence and starlikeness, Colloq. Math., 47 (1982), 309–314.
Some Inequalities Associated with a Linear Operator Defined
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[9] H.M. SRIVASTAVA AND S. OWA (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992.